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Network analysis of students’ conceptual understanding of mathematical expressions for probability in upper-division quantum mechanics

William D. Riihiluoma1, Zeynep Topdemir2, and John R. Thompson3,4

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Phys. Rev. Phys. Educ. Res. 20, 020102 – Published 15 July, 2024

DOI: https://doi.org/10.1103/PhysRevPhysEducRes.20.020102

Abstract

One expected outcome of physics instruction is for students to be capable of relating physical concepts to multiple mathematical representations. In quantum mechanics (QM), students are asked to work across multiple symbolic notations, including some they have not previously encountered. To investigate student understanding of the relationships between expressions used in these various notations, a survey was developed and distributed to students at six different institutions. All of the courses studied were structured as “spins-first,” in which the course begins with spin-1/2 systems and Dirac notation before transitioning to include continuous systems and wave function notation. Network analysis techniques such as community detection methods were used to investigate conceptual connections between commonly used expressions in upper-division QM courses. Our findings suggest that, for spins-first students, Dirac bras and kets share a stronger identity with vectorlike concepts than are associated with quantum state or wave function concepts. This work represents a novel way of using well-developed network analysis techniques and suggests such techniques could be used for other purposes as well.

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References (49)

  1. O. Uhden, R. Karam, M. Pietrocola, and G. Pospiech, Modelling mathematical reasoning in physics education, Sci. Educ. (Dordrecht) 21, 485 (2012).
  2. C. Singh, Student understanding of quantum mechanics, Am. J. Phys. 69, 885 (2001).
  3. C. Singh and E. Marshman, Review of student difficulties in upper-level quantum mechanics, Phys. Rev. ST Phys. Educ. Res. 11, 020117 (2015).
  4. C. Singh, Student understanding of quantum mechanics at the beginning of graduate instruction, Am. J. Phys. 76, 277 (2008).
  5. G. Passante, B. P. Schermerhorn, S. J. Pollock, and H. R. Sadaghiani, Time evolution in quantum systems: A closer look at student understanding, Eur. J. Phys. 41, 015705 (2020).
  6. P. J. Emigh, G. Passante, and P. S. Shaffer, Student understanding of time dependence in quantum mechanics, Phys. Rev. ST Phys. Educ. Res. 11, 020112 (2015).
  7. G. Karakok, Making connections among representations of eigenvector: What sort of a beast is it?, ZDM Int. J. Math. Educ. 51, 1141 (2019).
  8. M. Wawro, K. Watson, and M. Zandieh, Student understanding of linear combinations of eigenvectors, ZDM Int. J. Math. Educ. 51, 1111 (2019).
  9. M. O. J. Thomas and S. Stewart, Eigenvalues and eigenvectors: Embodied, symbolic and formal thinking, Math. Educ. Res. J. 23, 275 (2011).
  10. M. Wawro, K. Watson, and W. Christensen, Meta-representational competence with linear algebra in quantum mechanics, in Proceedings of CERME 10 (2017).
  11. A. Pina, Z. Topdemir, and J. R. Thompson, Student understanding of eigenvalue equations in quantum mechanics: Symbolic forms, in Proceedings of 25th RUME (2023).
  12. K. S. Serbin and M. Wawro, The inextricability of students’ mathematical and physical reasoning in quantum mechanics problems, Int. J. Res. Undergrad. Math. Educ. 10, 57 (2022).
  13. M. Wawro, K. Watson, and W. Christensen, Students’ metarepresentational competence with matrix notation and Dirac notation in quantum mechanics, Phys. Rev. Phys. Educ. Res. 16, 020112 (2020).
  14. B. P. Schermerhorn, G. Passante, H. R. Sadaghiani, and S. J. Pollock, Exploring student preferences when calculating expectation values using a computational features framework, Phys. Rev. Phys. Educ. Res. 15, 020144 (2019).
  15. E. Gire and E. Price, Structural features of algebraic quantum notations, Phys. Rev. ST Phys. Educ. Res. 11, 020109 (2015).
  16. D. J. Griffiths, Introduction to Quantum Mechanics, 1st ed. (Pearson, Upper Saddle River, NJ, 1995).
  17. D. H. McIntyre, Quantum Mechanics: A Paradigms Approach (Pearson, San Francisco, CA, 2012).
  18. J. S. Townsend, A Modern Approach to Quantum Mechanics (University Science Books, Mill Valley, CA, 2000).
  19. J. Stewart, J. Hansen, and E. Burkholder, Visualizing and predicting the path to an undergraduate physics degree at two different institutions, Phys. Rev. Phys. Educ. Res. 18, 020117 (2022).
  20. M. E. J. Newman, Networks: An Introduction, 1st ed. (Oxford University Press, New York, 2010).
  21. S. Thomas, Ties that bind: A social network approach to understanding student integration and persistence, J. Higher Educ. 71, 591 (2000).
  22. P. S. Smith, M. L. Hayes, and K. M. Lyons, The ecology of instructional teacher leadership, J. Math. Behav. 46, 267 (2017).
  23. M. Hopkins, D. Ozimek, and T. M. Sweet, Mathematics coaching and instructional reform: Individual and collective change, J. Math. Behav. 46, 215 (2017).
  24. J. Wells, R. Henderson, J. Stewart, G. Stewart, J. Yang, and A. Traxler, Exploring the structure of misconceptions in the Force Concept Inventory with modified module analysis, Phys. Rev. Phys. Educ. Res. 15, 020122 (2019).
  25. J. Wells, R. Henderson, A. Traxler, P. Miller, and J. Stewart, Exploring the structure of misconceptions in the Force and Motion Conceptual Evaluation with modified module analysis, Phys. Rev. Phys. Educ. Res. 16, 010121 (2020).
  26. J. Yang, J. Wells, R. Henderson, E. Christman, G. Stewart, and J. Stewart, Extending modified module analysis to include correct responses: Analysis of the Force Concept Inventory, Phys. Rev. Phys. Educ. Res. 16, 010124 (2020).
  27. C. Wheatley, J. Wells, R. Henderson, and J. Stewart, Applying module analysis to the conceptual survey of electricity and magnetism, Phys. Rev. Phys. Educ. Res. 17, 010102 (2021).
  28. E. Brewe, J. Bruun, and I. G. Bearden, Using module analysis for multiple choice responses: A new method applied to Force Concept Inventory data, Phys. Rev. Phys. Educ. Res. 12, 020131 (2016).
  29. J. Wells, H. Sadaghiani, B. P. Schermerhorn, S. Pollock, and G. Passante, Deeper look at question categories, concepts, and context covered: modified module analysis of quantum mechanics concept assessment, Phys. Rev. Phys. Educ. Res. 17, 020113 (2021).
  30. T. Wan, P. J. Emigh, and P. S. Shaffer, Investigating how students relate inner products and quantum probabilities, Phys. Rev. Phys. Educ. Res. 15, 010117 (2019).
  31. A. Kohnle and G. Passante, Characterizing representational learning: A combined simulation and tutorial on perturbation theory, Phys. Rev. Phys. Educ. Res. 13, 020131 (2017).
  32. R. P. Dalka, D. Sachmpazidi, C. Henderson, and J. P. Zwolak, Network analysis approach to Likert-style surveys, Phys. Rev. Phys. Educ. Res. 18, 020113 (2022).
  33. M. Bodin, Mapping university students’ epistemic framing of computational physics using network analysis, Phys. Rev. ST Phys. Educ. Res. 8, 010115 (2012).
  34. E. Brewe, L. Kramer, and V. Sawtelle, Investigating student communities with network analysis of interactions in a physics learning center, Phys. Rev. ST Phys. Educ. Res. 8, 010101 (2012).
  35. K. Commeford, E. Brewe, and A. Traxler, Characterizing active learning environments in physics using network analysis and classroom observations, Phys. Rev. Phys. Educ. Res. 17, 020136 (2021).
  36. M. Sundstrom, A. Schang, A. B. Heim, and N. G. Holmes, Understanding interaction network formation across instructional contexts in remote physics courses, Phys. Rev. Phys. Educ. Res. 18, 020141 (2022).
  37. M. E. J. Newman, Finding community structure in networks using the eigenvectors of matrices, Phys. Rev. E 74, 036104 (2006).
  38. M. E. J. Newman, Modularity and community structure in networks, Proc. Natl. Acad. Sci. U.S.A. 103, 8577 (2006).
  39. R. P. Springuel, M. C. Wittmann, and J. R. Thompson, Applying clustering to statistical analysis of student reasoning about two-dimensional kinematics, Phys. Rev. ST Phys. Educ. Res. 3, 020107 (2007).
  40. M. E. J. Newman and M. Girvan, Finding and evaluating community structure in networks, Phys. Rev. E 69, 026113 (2004).
  41. S. Fortunato and M. Barthélemy, Resolution limit in community detection, Proc. Natl. Acad. Sci. U.S.A. 104, 36 (2007).
  42. S. Fortunato, Community detection in graphs, Phys. Rep. 486, 75 (2010).
  43. M. E. J. Newman, Detecting community structure in networks, Eur. Phys. J. B 38, 321 (2004).
  44. M. E. J. Newman and M. Girvan, Finding and evaluating community structure in networks, Phys. Rev. E 69, 026113 (2004).
  45. M. E. J. Newman, Analysis of weighted networks, Phys. Rev. E 70, 056131 (2004).
  46. B. Efron and R. J. Tibshirani, An Introduction to the Bootstrap (Chapman & Hall, New York, 1993).
  47. J. C. Speirs, M. R. Stetzer, and B. A. Lindsey, Utilizing network analysis to explore student qualitative inferential reasoning chains, Phys. Rev. Phys. Educ. Res. 20, 010147 (2024).
  48. B. L. Sherin, How students understand physics equations, Cognit. Instr. 19, 479 (2001).
  49. B. P. Schermerhorn and J. R. Thompson, Making context explicit in equation construction and interpretation: Symbolic blending, Phys. Rev. Phys. Educ. Res. 19, 020149 (2023).

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Phys. Rev. Phys. Educ. Res. 20, 020102– Published 15 July, 2024
  • Received 21 February 2024
  • Accepted 1 July 2024
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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